
where \(q_t = \nabla \widetilde{\mathcal {W}}_2^2\big (\nu ^{\mathcal {G}_1, g} , \nu ^{\mathcal {G}_2, g}_{P_t} \big ) - \epsilon \nabla H(P_t)\) , the symbol \(\odot \)  is the Haddamard (pointwise) product between matrices, \(\nabla \)  represents the gradient operator, \(\alpha \)  is the step size and \(\epsilon \)  is the entropic regularisation parameter.
The KL projection \(\mathcal {P}_{\mathcal {C}_{\rm fuzzy}}^{\rm KL}\)  can be computed through the Sinkhorn operator \(\mathcal {S}_\tau \)  as [1]}, [2]}
\(\mathcal {P}_{\mathcal {C}_{\rm fuzzy}}^{\rm KL}(P) = \mathcal {S}_\tau (-\tau \log {P}),\) 
